Constructing classical realizability models of Zermelo-Fraenkel set theory

Transcription

1 Constructing classical realizability models of Zermelo-Fraenkel set theory Alexandre Miquel Plume team LIP/ENS Lyon June 5th, 2012 Réalisabilité à Chambéry

2 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

3 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

4 Why ZF ε? A similar difficulty occurs in the construction of a forcing model of ZF [Cohen 63] a Boolean-valued model of ZF [Scott, Solovay, Vopěnka] a realizability model of IZF [Myhill-Friedman 73, McCarty 84] a classical realizability model of ZF [Krivine 00] which is the interpretation of the axiom of extensionality : x y [x = y z (z x z y) The reason is that in these models, sets cannot be given a canonical representation need some extensional collapse (A similar problem occurs in CS when manipulating sets) Most authors solve the problem in the model, when defining the interpretation of extensional equality and membership Krivine proposes to address the problem in the syntax, using a non extensional presentation of ZF called ZF ε (= assembly language for ZF)

5 The language of ZF ε Formulas φ, ψ ::= x ε/ y x / y x y φ ψ x φ Abbreviations : φ φ φ ψ (φ ψ ) φ ψ φ ψ φ ψ (φ ψ) (ψ φ) x ε y (x ε/ y) x y (x / y) x y x y y x x {φ 1 & & φ n} x (φ 1 φ n ) ( x ε a) φ x (x ε a φ) ( x ε a) φ x {x ε a & φ} ( x a) φ x (x a φ) ( x a) φ x {x a & φ} A formula φ is extensional if it does not contain ε/ Formulas x y, x y, x y are extensional // x ε y is not. Extensional formulas are the formulas of ZF

6 The axioms of ZF ε Extensionality x y (x y ( z ε y) x z) Foundation Comprehension x y (x y ( z ε x) z y) z [ x (( y ε x)φ(y, z) φ(x, z)) x φ(x, z)] z a b x (x ε b x ε a φ(x, z)) Pairing a b c {a ε c & b ε c} Union a b ( x ε a) ( y ε x) y ε b Powerset a b x ( y ε b) z (z ε y z ε x z ε a) Collection Infinity z a b ( x ε a) [ y φ(x, y, z) ( y ε b) φ(x, y, z)] z a b {a ε b & ( x ε b) ( y φ(x, y, z) ( y ε b) φ(x, y, z))} Proofs formalized in natural deduction + Peirce s law

7 The extensional relations, and (1/2) Extensionality axioms define and by mutual induction x y ( y ε y) x y ( y ε y) {x y & y x } x y ( x ε x) x y ( x ε x) ( y ε y) {x y & y x } Foundation scheme expresses that ε is well-founded : z [ x (( y ε x)φ(y, z) φ(x, z)) x φ(x, z)] Combining Extensionality with Foundation, we get : Reflexivity : ZF ε x (x x) Induction hypothesis : φ(x) x x Consequences : ZF ε x (x x) ZF ε x y (x ε y x y)

8 The extensional relations, and (2/2) From Extensionality, we have : x y ( x ε x) ( y ε y) {x y & y x } Combined with Foundation again, we get : Transitivity : ZF ε x y z (x y y z x z) Induction hypothesis : φ(x) y z (x y y z x z) y z (z y y x z x) So that : Inclusion x y is a preorder Extensional equality x y is the associated equivalence relation Extensional (ZF) definitions of and are then derivable : ZF ε ZF ε x y [x y z (z x z y)] x y [x y z (z x z y)]

9 Extensional peeling We can now derive that is compatible with the two primitive extensional predicates / and : Extensional peeling ZF ε x y z (x y x / z y / z) ZF ε x y z (x y z / x z / y) ZF ε x y z (x y x z y z) ZF ε x y z (x y z x z y) For any extensional formula φ(x, z) : ZF ε z x y [x y (φ(x, z) φ(y, z))] Proof : by structural induction on φ(x, z) Remarks : Proof structurally depends on φ(x, z) non parametric Only holds when φ(x, z) is extensional. Counter-example : x y (x ε z y ε z)

10 Consequences of extensional peeling Extensional peeling is the tool to derive the usual extensional axioms of ZF from their intensional formulation in ZF ε. But schemes need to be restricted to extensional formulas (as in ZF) In ZF ε, (intensional) Foundation and Comprehension schemes z [ x (( y ε x)φ(y, z) φ(x, z)) x φ(x, z)] z a b x (x ε b x ε a φ(x, z)) hold for any formula φ(x, z) (may contain ε) Combined with extensional peeling, we get Foundation & Comprehension : ZF formulation ZF ε ZF ε z [ x (( y x)φ(y, z) φ(x, z)) x φ(x, z)] z a b x (x b x a φ(x, z)) for any extensional formula φ(x, z) (cannot contain ε)

11 Leibniz equality and intensional peeling Leibniz equality is definable in ZF ε : x = y z (x ε/ z y ε/ z) (Could replace ε/ by ε) Thanks to (intensional) Comprehension, we get : Intensional peeling For any formula φ(x, z) : ZF ε z x y [x = y (φ(x, z) φ(y, z))] Proof : We only need to prove x = y (φ(y, z) φ(x, z)). (For the converse direction : replace φ(x, z) by φ(x, z).) Assume x = y and φ(y, z). From Pairing, there exists u such that y ε u. From Comprehension, there exists u such that x (x ε u x ε u φ(x, z)). By construction, we have y ε u (since y ε u and φ(y, z)). Since x = y, we get x ε u (by contraposition). Therefore : x ε u and φ(x, z). Remarks : Proof does not structurally depend on φ(x, z) parametric This property holds for any formula φ(x, z).

12 Strong inclusion, strong equivalence Let x y z (z ε x z ε y) x y z (z ε x z ε y) ( x y y x) Remarks : x y is a preorder, stronger than x y x y is the associated equivalence x y weaker than x = y, stronger than x y (None of the converse implications is derivable) Going back to Comprehension : z a b x (x ε b x ε a φ(x, z)) The set b = {x ε a : φ(x)} is unique up to (and thus up to ), but not up to = (Leibniz equality)

13 Pairing and union In ZF ε, the (intensional) axioms of Pairing and Union only give upper approximations of the desired sets : a b c {a ε c & b ε c} a b ( x ε a) ( y ε x) y ε b Cutting them by Comprehension, we get what we expect : ZF ε a b c x (x ε c x = a x = b) ZF ε a b x (x ε b ( y ε a) x ε y) Note that b and c are unique up to strong equivalence. And by extensional peeling, we get : Pairing and Union : ZF formulation ZF ε a b c x (x c x a x b) ZF ε a b x (x b ( y a) x y)

14 Powerset In ZF ε, the (intensional) Powerset axiom only gives an upper approximation of the desired set : a b x ( y ε b) z (z ε y z ε x z ε a) Intuitively : b contains a copy of all sets of the form x a Cutting b with Comprehension, we get : ZF ε a b {( x ε b ) x a & x (x a ( x ε b ) x x )} Here, b is unique up to, but not up to. Cannot do better, since {x : x a} is a proper class in realizability models. And by extensional peeling, we get : Powerset : ZF formulation ZF ε a b x (x b x a)

15 Collection and Infinity ZF ε comes with Collection and Infinity schemes : z a b ( x ε a) [ y φ(x, y, z) ( y ε b) φ(x, y, z)] z a b {a ε b & ( x ε b) ( y φ(x, y, z) ( y ε b) φ(x, y, z))} for every formula φ(x, y, z) Collection and Infinity schemes : extensional formulation ZF ε ZF ε z a b ( x ε a) [ y φ(x, y, z) ( y b) φ(x, y, z)] z a b {a b & ( x b) ( y φ(x, y, z) ( y b) φ(x, y, z))} for every extensional formula φ(x, y, z) In general, Collection is stronger than Replacement but in ZF, they are equivalent due to Foundation Infinity scheme implies the existence of infinite sets and it is equivalent in presence of Collection

16 Conservativity All axioms of ZF are derivable in ZF ε : Proposition : ZF ε is an extension of ZF Collapsing ε and : For every formula φ of ZF ε, write φ the formula of ZF obtained by collapsing ε/ to / in φ. Proposition : If ZF ε φ, then ZF φ Therefore, if ZF is consistent, then none of the formulas x y (x y x ε/ y), x y (x y x y), etc. is derivable in ZF ε! (But they are realized... ) Theorem (Conservativity) ZF ε is a conservative extension of ZF (and thus equiconsistent) Proof : Assume ZF ε φ, where φ is extensional. Then ZF φ. But φ φ.

17 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

18 The λ c -calculus (1/2) Syntax Terms Stacks Processes t, u ::= x λx. t tu κ k π π ::= α t π p, q ::= t π (κ K) (α Π 0, t closed) (t closed) Syntax of the language is parameterized by A nonempty countable set K = {cc;...} of instructions A nonempty countable set Π 0 = {α;...} of stack constants A term is proof-like if it contains no k π (i.e. refers to no α Π 0 ) Notations : Λ = set of closed terms Π = set of stacks Λ Π = set of processes PL = set of closed proof-like terms ( Λ) Each natural number n ω is encoded as n = s n 0 ( PL) where 0 λxy. x and s λnxy. y (n x y)

19 The λ c -calculus (2/2) We assume that the set Λ Π comes with a preorder p p of evaluation satisfying the following rules : Krivine Abstract Machine (KAM) Push Grab Save Restore (+ reflexivity & transitivity) tu π t u π λx. t u π t{x := u} π cc u π u k π π k π u π u π Evaluation not defined but axiomatized. The preorder p p is another parameter of the calculus, just like the sets K and Π 0 Extensible machinery : can add extra instructions and rules (We shall see examples later) An instance of the λ c -calculus is defined by the triple (K, Π 0, )

20 Standard algebras Each classical realizability model (which is based on the λ-calculus) is parameterized by a set of processes Λ Π which is saturated, or closed under anti-evaluation (w.r.t. ) : If p p and p, then p Such a set is used as the pole of the model We call a standard algebra any pair A ((K, Π 0, ), ) formed by An instance (K, Π 0, ) of the λ c-calculus A saturated set Λ Π (i.e. the pole of the algebra A ) We shall first see how to build a realizability model M (A ) from an arbitrary standard algebra A. But this construction more generally works when A is an arbitrary realizability algebra (We shall see the general definition later)

21 The ground model M The whole construction is parameterized by : An arbitrary model M of ZFC, called the ground model An arbitrary standard algebra A M, which is taken as a point of the ground model M In what follows, we call a set any point of M We shall never consider sets outside M! We write ω M the set of natural numbers in M. Elements of ω are called the standard natural numbers 1 We consider the sets Λ, Π,, that are defined from A as points of the ground model M All set-theoretic notations (e.g. P(X ), {x : φ(x)}, etc.) are taken relatively to the ground model M Only formulas (of ZF ε ) live outside the ground model M 1. This is just a convention of terminology. The set ω might contain numbers that are non standard according to the external/ambient/intuitive/meta theory.

22 Building the model M (A ) of A -names By induction on α On ( M ), we define a set M α (A ) M α = P(M β Π) Note that : M (A ) 0 = M (A ) α+1 = P(M (A ) α Π) M (A ) α = β<α M (A ) β β<α (for α limit ordinal) by We write M (A ) = α M (A ) α the (proper) class of A -names Given a name a M (A ), we write dom(a) = {b : ( π Π) (b, π) a} (the domain of a) rk(a) the smallest α On s.t. a M (A ) α (the rank of a)

23 Structure of the interpretation Variables x 1,..., x n,... of the language of ZF ε are interpreted as names a 1,..., a n,... M (A ) We call a formula with parameters in M (A ) any formula of ZF ε enriched with constants taken in M (A ) : φ(x 1,..., x k ) + a 1,..., a k M (A ) φ(a 1,..., a k ) Formulas with parameters in M (A ) constitute the language of the realizability model M (A ) Closed formulas φ with parameters in M (A ) are interpreted as two sets (i.e. points of M ) : A falsity value φ P(Π) A truth value φ P(Λ), defined by orthogonality : φ = φ = {t Λ : ( π φ ) (t π )}

24 Interpreting formulas Given a closed formula φ with parameters in M (A ) : Falsity value φ P(Π) defined by induction on the size of φ a ε/ b, a / b, a b = (postponed) = = Π φ ψ = φ ψ = {t π : t φ, π ψ } x φ(x) = φ(a) = {π Π : ( a M (A ) ) π φ(a) } a M (A ) Truth value φ P(Λ) defined by orthogonality φ = φ = {t Λ : ( π φ ) (t π )} Notations : t φ t φ M (A ) φ θ φ for some θ PL φ PL (t realizes φ) (φ is realized)

25 Anatomy of the interpretation Denotation of units : Falsity value = Truth value = = Λ = Π = Π (by definition) (by orthogonality) Denotation of universal quantification : Falsity value : x φ(x) = φ(a) a M (A ) Truth value : x φ(x) = φ(a) a M (A ) (by definition) (by orthogonality) Denotation of implication : Falsity value : φ ψ = φ ψ (by definition) Truth value : φ ψ φ ψ (by orthogonality) writing φ ψ = {t Λ : u φ tu ψ } (realizability arrow) 1 Converse inclusion does not hold in general, unless closed under Push 2 In all cases : If t φ ψ, then λx. tx φ ψ (η-expansion)

26 Adequacy Deduction/typing rules Γ x : φ (x:φ) Γ Γ t : FV (t) dom(γ) Γ t : Γ t : φ Γ, x : φ t : ψ Γ λx. t : φ ψ Γ t : φ Γ t : x φ x / FV (Γ) Γ Γ t : φ ψ Γ u : φ Γ tu : ψ t : x φ Γ t : φ{x := e} Γ cc : ((φ ψ) φ) φ (e first-order term) Adequacy Given : a derivable judgment x 1 : φ 1,..., x n : φ n t : φ a valuation ρ (in M (A ) ) closing φ 1,..., φ n, φ realizers u 1 φ 1 [ρ],..., u n φ n [ρ] We have : t{x 1 := u 1 ;... ; x n := u n } φ[ρ]

27 Interpreting intensional membership Interpretation of ε/ reminiscent from forcing in ZF [Cohen 63] and intuitionistic realizability in IZF [Myhill-Friedman 73, McCarty 84] In forcing / int. realizability, a name a M (C) is a set of pairs (b, p) where p C is a certificate witnessing that b ε a : (b, p) a means : p forces/realizes b ε a hence : b ε a = {p C : (b, p) a} In forcing : p is a forcing condition In intuitionistic realizability : p is a realizer But in classical realizability, we use refutations (i.e. stacks) instead : (b, π) a means π refutes b ε/ a hence : b ε/ a = {π Π : (b, π) a} π b ε/ a implies k π b ε a ( b ε/ a) b ε/ a = = as soon as b / dom(a)

28 Interpreting atomic formulas Interpretation of a ε/ a, a b and a / b (a, a, b M (A ) ) a ε/ a = {π Π : (a, π) a} a b = a / b a ε/ a a / b = a dom(a) b dom(b) a b b a b ε/ b Def. of a ε/ a is primitive (i.e. non recursive) Def. of a b and a / b is mutually recursive Def. of a b calls a / b Def. of a / b calls a b and b a for all a dom(a) for all b dom(b) Hence the definition of a b for a, b M α (A ) recursively calls a b for a, b M (A ) β where β < α

29 The interpretation of Since c ε/ a = as soon as c / dom(a) : a b = c / b c ε/ a = c dom(a) c M (A ) c / b c ε/ a = z (z / b z ε/ a) Hence the atomic formula x y has the very same semantics as the formula z (z / y z ε/ x) By adequacy, we can build θ PL such that (Exercise : find θ) θ x y [ z (z / y z ε/ x) ( z ε x) z y] Realizing Extensionality for : θ x y (x y ( z ε x) z y)

30 The interpretation of / Since c ε/ b = as soon as c / dom(b) : a / b = a c c a c ε/ b = c dom(b) c M (A ) a c c a c ε/ b = z (a z z a z ε/ b) Hence the atomic formula x / y has the very same semantics as the formula z (x z z x z ε/ y) By adequacy, we can build θ PL such that (Exercise : find θ ) θ x y [ z (x z z x z ε/ y) ( z ε y) x z] Realizing Extensionality for : θ x y (x y ( z ε y) x z)

31 Discriminating ε and Let = and = { } In the case where, we have : Π = Π Π But both names and represent the empty set : 1 θ x (x ε/ ) (θ PL arbitrary) 2 I x (x ε/ ) 3 Therefore : M (A ) Writing a = { } Π, we get : 1 I ε a and θ ε/ a (θ PL arbitrary) 2 Therefore : M (A ) 3 Moreover : M (A ) a (since M (A ) )

32 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

33 Realizing the axioms of ZF ε For every axiom φ of ZF ε, we want to show that : There is θ PL such that Which we write : M (A ) φ We have already shown that : Realizing Extensionality θ φ M (A ) x y (x y ( z ε y) x z) M (A ) x y (x y ( z ε x) z y) We now need to realize the following : Foundation scheme Comprehension scheme Pairing and Union axioms Powerset axiom Collection & Infinity schemes (we shall only consider Collection)

34 Realizing Foundation Consider Turing s fixpoint combinator : Y (λyf. f (y y f )) (λyf. f (y y f )) We have : Y t π t (Y t) π (t Λ, π Π) Proposition For any formula ψ(x) with parameters in M (A ), we have : Y x ( y (ψ(y) y ε/ x) ψ(x)) x ψ(x) Proof : We show that Y x ( y (ψ(y) y ε/ x) ψ(x)) ψ(a) for all a M (A ), by induction on rk(a). Realizing foundation For any formula φ(x, z), we have : M (A ) z [ x (( y ε x) φ(y, z) φ(x, z)) x φ(x, z)]

35 Realizing witnessed existential formulas Lemma Let φ(x 1,..., x n, y) be a formula and θ PL such that : ( a 1,..., a n M (A ) ) ( b M (A ) ) θ φ(a 1,..., a n, b) Then : λz. z θ x 1 x n y φ(x 1,..., x n, y) Lemma More generally : Given k formulas φ 1( x, y),..., φ k ( x, y) k terms θ 1,..., θ k PL such that : ( x x 1,..., x n) ( a M (A ) ) ( b M (A ) ) (θ 1 φ 1( a, b) θ k φ k ( a, b)) Then : λz. z θ 1 θ k x y {φ 1( x, y) & & φ k ( x, y)}

36 Realizing Comprehension (1/2) Given a name a M (A ) and a formula φ(x) (with params in M (A ) ) Let : b = By construction, we have : dom(b) dom(a) c dom(a) {c} φ(c) c ε/ a c ε/ b = φ(c) c ε/ a for all c M (A ) (Since c ε/ b = = φ(c) c ε/ a as soon as c / dom(a)) This means that : x ε/ b has the same semantics as φ(x) x ε/ a x ε b x ε/ b has the same semantics as (φ(x) x ε/ a)

37 Realizing Comprehension (2/2) Let θ 1 and θ 2 be proof-like terms such that : θ 1 θ 2 x [ (φ(x) x ε/ a) x ε a φ(x)] x [x ε a φ(x) (φ(x) x ε/ a)] Since x ε b has the same semantics as (φ(x) x ε/ a) : θ 1 x [x ε b x ε a φ(x)] θ 2 x [x ε a φ(x) x ε b] λu. u θ 1 θ 2 x [x ε b x ε a φ(x)] Hence (by Lemma) : Realizing Comprehension For every formula φ( z, x) : λz. z (λu. u θ 1 θ 2 ) z a b x (x ε b x ε a φ(x, z))

38 Realizing Pairing Given a, b M (A ), let c = {a; b} Π We have a ε/ c = b ε/ c =, hence : Hence (by Lemma) : Realizing Pairing I a ε c ( a ε/ c) I b ε c ( b ε/ c) λz. z I I a b c {a ε c & b ε c}

39 Realizing Union Given a M (A ), let b = a dom(a) a Lemma For all a, a M (A ) : a ε/ b a ε/ a a ε/ a a ε/ a Proof : We notice that a ε/ a a ε/ b as soon as a dom(a). Hence I x y ((y ε/ x x ε/ a) (y ε/ b x ε/ a)) so we can find θ PL such that : Therefore : Realizing Union θ x y (x ε a y ε x y ε b) λz. z θ a b ( x ε a) ( y ε x) y ε b

40 Realizing Powerset Given a M (A ), let b = P(dom(a) Π) Π For every c M (A ), write : c a = We notice that : d dom(a) {d} d ε c d ε/ a 1 Formula z ε/ c a has the same semantics as z ε c z ε/ a. Hence there is θ PL such that : θ z (z ε c a z ε c z ε a) 2 dom(c a ) P(dom(a) Π), hence c a ε/ b =, and thus : I c a ε b Therefore : Realizing Powerset λz. z (λz. z I θ) a b x ( y ε b) z (z ε y z ε x z ε a)

41 Realizing Collection Let φ(x, y) a formula with parameters in M (A ) and a M (A ) Using Collection in M, consider a set B such that : ( c dom(a)) ( t Λ) [ d (d M (A ) t φ(c, d)) ( d B) (d M (A ) t φ(c, d))] (Wlog, we can assume that B M (A ) ) Writing b = B Π, we have : Lemma For all c M (A ) : y (φ(c, y) x ε/ a) y (φ(c, y) y ε/ b) Hence I x [ y (φ(x, y) y ε/ b) y (φ(x, y) x ε/ a)] so there is θ PL s.t. : θ ( x ε a) [ y φ(x, y) ( y ε b) φ(x, y)] Realizing Collection For every formula φ(x, y, z) : λz. z θ z a b ( x ε a) [ y φ(x, y, z) ( y ε b) φ(x, y, z)]

42 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

43 Adding function symbols (1/2) It is often convenient to enrich the language of ZF ε with a k-ary function symbol f interpreted as a k-ary class function f : M (A ) M }{{ (A } ) M (A ) k We say that f is extensional when M (A ) x y ( x y f ( x) f ( y)) Beware : This is usually not the case! But in all cases, we have M (A ) (due to intensional peeling) x y ( x = y f ( x) = f ( y))

44 Adding function symbols (2/2) Example : Consider the successor function s( ), that is defined for all a M (A ) by s(a) = {(b, 0 π) : (b, π) dom(a)} {(a, 1 π) : π Π} Intensional/extensional characterization of s 1 M (A ) x y (y ε s(x) y ε x y = x) 2 M (A ) x y (y s(x) y x y x) 3 The successor function s is extensional Actually, this function is intensionally injective : M (A ) x y (s(x) = s(y) x = y) Proof : Consider a function p( ) ( predecessor ) such that p(s(a)) = a for all a M (A )

45 Constructing the set ω of natural numbers Let 0 = and ñ + 1 = s(ñ) (for all n ω) Put ω = {(ñ, n π) : n ω, π Π} Intensional properties of ω M (A ) y (y ε/ 0) M (A ) x y (y ε s(x) y ε x y = x) M (A ) M (A ) M (A ) 0 ε ω ( x ε ω) s(x) ε ω φ( 0) ( x ε ω) (φ(x) φ(s(x))) ( x ε ω) φ(x) where φ(x) is any formula with parameters in M (A ) Remark : This implementation of ω provides a canonical intensional representation of natural numbers : M (A ) ( x ε ω) ( y ε ω) (x y x = y)

46 The type of natural numbers : ωג Recall that : ω = {( p, p π) : p ω, π Π} and put : ωג = {( p, π) : p ω, π Π} Π} nג = {( p, π) : p < n, π From the definition, we have : M (A ) ω ωג Distinction between (intensional) elements of ω and of ωג is the same as between natural numbers and individuals in 2nd-order logic Krivine showed that in some models (such as the threads model) : Inclusion ω ωג is strict ωג is (intensionally) not denumerable Subsets nג ωג have amazing (intensional) cardinality properties However, the set ωג is extensionally equal to ω : M (A ) ωג ω

47 The non extensional axiom of choice (NEAC) (1/2) Lemma Add an instruction quote with the rule quote t u π u n t π where n t is the index of t according to a fixed bijection n t n from ω to Λ Let φ(x 1,..., x k, y) be a formula Consider the (k + 1)-ary function symbol f φ interpreted by 2 f φ (a 1,..., a k, ñ) = some b M (A ) s.t. t n φ(a 1,..., a k, b) if there is such a name b f φ (a 1,..., a k, b) = in all the other cases λxy. quote y (x y) x [ n (φ( x, f φ ( x, n)) n ε/ ω) y φ( x, y)] 2. Assuming that M interprets the choice principle (= conservative ext. of ZFC)

48 The non extensional axiom of choice (NEAC) (2/2) M (A ) x [ n (φ( x, f φ ( x, n)) n ε/ ω) y φ( x, y)] Taking the contrapositive, we get : Non extensional axiom of choice (NEAC) M (A ) x [ y φ( x, y) ( n ε ω) φ( x, f φ ( x, n))] Remarks (f φ ( a, n)) n ω is a denumerable sequence of potential witnesses of the existential formula y φ( a, y) The function f φ is not extensional in general, even in the case where the formula φ is extensional Nevertheless, NEAC is strong enough to imply the axiom of dependent choices (DC)

49 Alternative formulation of NEAC (1/3) NEAC : M (A ) x [ y φ( x, y) ( n ε ω) φ( x, f φ ( x, n))] Consider the abbreviations : ψ 0( x, n) φ( x, f φ ( x, n)) ψ 1( x, n) ( m ε ω) (ψ 0( x, m) m ε/ n) ( there is witness at index n ) ( no witness below index n ) From the minimum principle, we get : M (A ) x [ y φ( x, y) ( n ε ω) {ψ 0 ( x, n) & ψ 1 ( x, n)}] Idea : Introduce a k-ary function h φ such that h φ ( x) f φ ( x, n), where n is the smallest index s.t. φ( x, f φ ( x, n))

50 Alternative formulation of NEAC (2/3) For all a = a 1,..., a k M (A ), let : h φ ( a) = {b} S a,b b D a where : D a = dom(f φ ( a, ñ)) n ω S a,b = ( n ε ω) (ψ 0( a, n) ψ 1( a, n) b ε/ f φ ( a, n)) By def. of h φ ( a), we have for all b M (A ) : b ε/ h φ ( a) = ( n ε ω) (ψ 0( a, n) ψ 1( a, n) b ε/ f φ ( a, n)) Therefore : M (A ) x z [z ε h φ ( x) ( n ε ω) {ψ 0( x, n) & ψ 1( x, n) & z ε f φ ( x, n)}]

51 Alternative formulation of NEAC (3/3) We have shown : M (A ) M (A ) x [ y φ( x, y) ( n ε ω) {ψ 0 ( x, n) & ψ 1 ( x, n)}] x z [z ε h φ ( x) ( n ε ω) {ψ 0 ( x, n) & ψ 1 ( x, n) & z ε f φ ( x, n)}] Combining these results, we get : Alternative formulation of NEAC 1 For any formula φ( x, y) : M (A ) x [ y φ( x, y) y {y h φ ( x) & φ( x, y)}] 2 If moreover the formula φ( x, y) is extensional : M (A ) x [ y φ( x, y) φ( x, h φ ( x))] Beware! The function h φ is in general non extensional, even when the formula φ( x, y) is But h φ can be used in Comprehension, Collection, etc.

52 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

53 From the λ c -calculus to realizability algebras Realizability algebras [Krivine 10] Same idea as PCAs (or OPCAs), but for classical realizability Each realizability algebra A contains a pole, and defines a classical realizability model M (A ) of ZF ε (from a ground model M ) Construction of M (A ) is the same as in the standard case Realizability algebras may be built from The λ c-calculus or Parigot s λµ-calculus Curien-Herbelin s λµ-calculus Any complete Boolean algebra (CBN or CBV) Realizability algebras can combine (standard) classical realizability with Cohen forcing iterated forcing [Krivine 10] Slogan : classical realizability = non commutative forcing

54 Realizability algebras (1/2) [Krivine 10] Some terminology (where A is a fixed set) : Proof-term λ-term with cc Proof-terms t, u ::= x λx. t tu cc A-environment finite association list σ (Var A) Notations : σ x 1 := a 1,..., x n := a n dom(σ) = {x 1;... ; x n} cod(σ) = {a 1;... ; a n} Environments are ordered, variables may be bound several times Compilation function into A function (t, σ) t[σ] taking : proof-term t + A-environment σ closing t, returning : element t[σ] A

55 Realizability algebras (2/2) [Krivine 10] Definition A realizability algebra A is given by : 3 sets Λ (A -terms), Π (A -stacks), Λ Π (A -processes) 3 functions ( ) : Λ Π Π, ( ) : Λ Π Λ Π, (k ) : Π Λ A compilation function (t, σ) t[σ] into the set Λ of A -terms A subset PL Λ (of proof-like A -terms) such that for all (t, σ) : If cod(σ) PL, then t[σ] PL (FV (t) dom(σ)) A set of A -processes Λ Π (the pole) such that : σ(x) π implies x[σ] π t[σ, x := a] π implies (λx. t)[σ] a π t[σ] u[σ] π implies (tu)[σ] π a k π π implies cc[σ] a π a π implies k π a π

56 Canonical example : the λ c -calculus Terms, stacks and processes Instructions Terms Stacks Processes κ ::= cc quote t, u ::= x λx. t tu κ k π π, π ::= α t π (α Π 0, t closed) p, q ::= t π (t closed) Λ, Π, Λ Π = sets of closed terms, stacks, processes Compilation t[σ] = substitution PL = set of closed terms containing no k π = any set of processes closed under anti-evaluation

57 Variant : the combinatory λ c -calculus (1/2) Terms, stacks and processes Instructions Terms Stacks Processes κ ::= I C B K W cc t, u ::= x κ tu k π π, π ::= α t π (α Π 0, t closed) p, q ::= t π (t closed) Krivine Abstract Machine (KAM) I K W C B Push Save Restore I t π t π K t u π t π W t u π t u u π C t u v π t v u π B t u v π t (uv) π tu π t u π cc u π u k π π k π u π u π

58 Variant : the combinatory λ c -calculus (2/2) Abstraction λ x. t is defined from binary abstraction λ x. t r : Definition of λ x. t r λ x. t r K (r t) λ x. x r r λ x. t 1t 2 r λ x. t 2 B r t 1 λ x. t 1t 2 r λ x. t 1 C (B r) t 2 λ x. t 1t 2 r W λ x. t 2 C λ x. t 1 B r (x / FV (t)) (x / FV (t 1 ), x FV (t 2 )) (x FV (t 1 ), x / FV (t 2 )) (x FV (t 1 ), x FV (t 2 )) Lemma For all t, u, r, π : λ x. t r u π r t{x := u} π Then we let : λ x. t λ x. t I Lemma For all t, u, π : λ x. t u π t{x := u} π Compilation function defined as expected, compiling λ as λ

59 Turning Boolean algebras into realizability algebras From a Boolean algebra B, we can build a realizability algebra A = (Λ, Π, Λ Π,..., ), letting : Λ = Π = Λ Π = B b 1 b 2 = b 1 b 2 = b 1b 2, k b = b PL = {1} t[σ] = σ(x) = {0} x FV (t) In the case where B is complete, the realizability model M (A ) is elementarily equivalent to the Boolean-valued model M (B) If B is not complete, then A automatically completes B

60 Plan 1 The theory ZF ε 2 The model M (A ) of A -names 3 Realizing the axioms of ZF ε 4 Realizing more axioms 5 Realizability algebras 6 Properties of the model M (A )

61 (blackboard)